Question

How do you find the maclaurin series expansion of `(1-x)^-2`?

Answer 1

`(1-x)^(-2)=d/dx((1-x)^(-1))=1+2x+3x^2+4x^3+5x^4+cdots` for `|x|<1`.

Explanation:

The quickest way is to differentiate the Maclaurin series expansion of `(1-x)^(-1)`, which itself can be found as the sum of a geometric series:

`(1-x)^(-1)=1/(1-x)=1+x+x^2+x^3+x^4+cdots` for `|x|<1`, so

`(1-x)^(-2)=d/dx((1-x)^(-1))=1+2x+3x^2+4x^3+5x^4+cdots` for `|x|<1`.